# Tag Info

3

subscript is a derivative with respect to this variable. $$u_{xx} = \frac{\partial^2u}{\partial x \partial x}$$

3

Answering the second question with a "standard" example of a function that is not of exponential order, but does have a Laplace transform in the region $\Re s>0$. Build a function out of spiky triangles $$\Delta_{H,A}(x)= \begin{cases} H-\frac{H^2}{A}|x|,&\ \text{if |x|\le A/H, and}\\ 0,&\ \text{otherwise.} \end{cases}$$ Here $H>0,A>0$ ...

3

For the first question, take something like $$f(t) = \sin(e^{t^2}).$$ Then $f$ is bounded (so in particular of exponential growth), but $$f'(t) = 2t e^{t^2} \cos( e^{t^2} )$$ is not of exponential growth. (Look at the values for $z=2\pi k$, $k \in \mathbb{Z}$.) For the second question, see Jyrki's answer.

2

EDIT: This proves the wrong thing; ignore this answer. If $f'$ is of exponential order, so is $f$. Consider the integral $$\int_a^x f'(t) dt$$ From the fundamental theorem of calculus, this differs from $f$ by at most a constant. If $|f(x)| ≤ |g(x)|$ for sufficiently large $x$, $\int_a^x |f(t)| dt ≤ \int_a^x |g(t)| dt$ for sufficiently large $x$ and $a$. $$... 2 It is a quotient space! \mathbb{R} here should be thought of as the one-dimensional subspace of L^2(\Omega) which consists of the (a.e.) constant functions. The norm given is the canonical norm on the quotient of a Banach space mod a closed subspace. This construction is discussed, for instance, in section III.4 of Conway's A Course in Functional ... 2 Let g be the restriction of \Phi to the boundary of the square. Your boundary data determines g up to a constant. You can then take this g and solve the corresponding Dirichlet problem; standard theory tells that solutions exist uniquely. Therefore you can conclude that solutions are unique up to shifting by constants. (The fact that shifting a ... 2 All given g don't involve \bar z very much: it appears to the first power, at most. So it makes sense to look for f of similar form:$$f(z) = \bar z f_1(z)+\bar z^2 f_2(z)\tag1$$where  f_1,f_2 are holomorphic. (More generally, one could consider \sum_{n=0}^N \bar z^n f_n(z), etc.) Applying the \partial/\partial\bar z operator to (1) and ... 2 Actually, it is more a way to easily remember the method (and I guess you are talking about separable differential equation here). We have$$\begin{align} \frac{\partial y}{\partial x}&=\frac{Q(y(x,z))}{P(x)}\\ \frac{\frac{\partial y}{\partial x}}{Q(y(x,z))} &=\frac{1}{P(x)}\\ \int \frac{\frac{\partial y}{\partial x}}{Q(y(x,z))}\text{d}x &= ...

2

I know I'm giving you bad news, but what you in fact have is this (where obviously $A_{jk}$ are block matrices): $$A_{11}x_{1}+A_{12}x_{2}+A_{13}x_{3}=b_{1}$$ $$A_{21}x_{1}+A_{22}x_{2}+A_{23}x_{3}=b_{2}$$ $$A_{31}x_{1}+A_{32}x_{2}+A_{33}x_{3}=b_{3}$$ Now...unfortunately if you don't have any other restriction I guess that's as far as you can get. If you ...

2

$U$ is some other normed vector space. In this case $L^2([0,T];U)$, sometimes lazily written as $L^2(0,T;U)$, consists of functions $f$ from $[0,T]$ to $U$ such that $\int_0^T \| f(t) \|^2 dt<\infty$, where $\| \cdot \|$ is the norm on $U$. This notation is used in, for instance, Partial Differential Equations by Evans. Most commonly the $U$ in question ...

2

sometimes a small transformation of a problem makes it easier to approach. here you may write the transformation as $yt=x$. then, differentiating wrt t we get (using the product rule) $$y+t\dot{y}=\dot{x}=f(y)$$ and the required answer is obtained by a simple algebraic manipulation

2

Smooth functions are dense in $L^2$, and you can pick an arbitrarily small (in whatever norm you like) smooth function with specified boundary values on $I$. So if $f_n \to f$, $f \in L^2$, you can replace $f_n$ with $f_n-g_n$, where say $\|g_n\|_{H^1} \leq \varepsilon$. So $f_n - g_n \to f$ in $L^2$, and $f_n-g_n$ is a smooth function that vanishes on the ...

1

If $y=\frac{x}{t}$, we have \begin{align} \dot{y} &= \frac{\dot{x}t-x}{t^{2}} \tag{derivative of a quotient}\\ &= \frac{f\left(\frac{x}{t}\right)t-x}{t^{2}} \tag{\dot{x}=f(x/t)}\\ &= \frac{f\left(\frac{x}{t}\right)t}{t^{2}}-\frac{x}{t}\cdot\frac{1}{t}\\ &= \frac{f(y)}{t} - y\cdot\frac{1}{t} \tag{since t\neq 0}\\ &= \frac{f(y)-y}{t} ... 1 Pick a point (x_0,t_0) with |x_0|>R+t_0. The initial values vanish on the ball B(x_0,|x_0|-R). By finite speed of propagation, the solution vanishes on the cone K built on top of this ball such that the t-time section of this cone is B(x_0,|x_0|-R-t)\times \{t\} for t<|x_0|-R. In particular, the t_0 section of K is ... 1 It seems like you need some further clarification. What you want to do is to fix a point x and a time t, and consider them as known. Now your solution is f(t,x) = E[ h(X_T) | x,t]where X_u is a stochastic process satisfying \begin{aligned} &\mathrm{d} X_u = 2u \, \mathrm{d}u + u^2\, \mathrm{d}W_u \quad \text{when} \quad t ... 1 The underlying problem is that the plate is subjected to distributed load with density f, and is fixed along the boundary in such a way that the boundary can neither mode nor rotate (clamping condition). The function u represents vertical displacement of the plate when it has reached an equilibrium. Since an equilibrium is being studied, no time is ... 1 You are confusing the multivariant and single-variant chain rules in your fomula. You don't get the multivariant version simply by replacing "d" with "\partial". f is function of two variables, not just one, and \zeta and \eta are functions of both x and y, not just one or the other. The chain rule for two variables is:\frac{\partial ...

1

$u$ is a two variable function $u(x,t)$, $$u_{xx}=\frac{\partial^2x}{\partial x^2}$$.

1

Hint: Just solve the equation using the standard separate of variable approach with initial condition as a sinusoidal function. Be sure you can generalize this approach to dealing with any other initial conditions using Fourier series.

1

If you apply maximum principle on $\Omega \setminus B_{\epsilon}(\mathbf{x}_0)$, there are two boundaries. As you have said Green function takes zero on $\partial \Omega$. For another boundary, note that $H$, being harmonic, is a twice continuously differentiable function in $\Omega$, so $|H(\mathbf{x}_0)|<\infty$, which implies that $\mathbf{x}_0$ is a ...

1

After the basic shock waves $x=t/2$ and $x=t+2$ (shown in red) are accounted for, the characteristics look like this. This picture is correct for times $t<1$, but needs two adjustments after that. Note that the solution within rarefaction wave is $u(x,t) = (x-1)/t$. Rarefaction catches up with shock on the right: $x=3$, $t=1$. From this point on, ...

1

Seek a solution of the form $$u(x,t)=\sum_{\lambda}C_{\lambda}(t)X_{\lambda}$$ where $X_{\lambda}$ are eigenfunctions of the operator $\frac{d^2}{dx^2}$ with homogeneous Neumann boundary conditions. We know the eigenfunctions are $1$ and $\cos(nx)$, $n \in \mathbb{N}$. Thus, we have $$u(x,t)=\sum_{n=0}^{\infty}C_n(t)\cos(nx)$$ Substituting the trial ...

1

These terms come from applying the product rule when multiplying out the operators: For example, the outer terms read as $$\left(\cos\theta\frac{\partial}{\partial r}\right)\left(-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)=-\cos\theta\frac{\partial}{\partial r}\left(\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)$$ You must then ...

1

If you want to avoid the measure theoretic framework, there is a quick and easy way to build a useful integral for Banach-valued functions. It's called the regulated integral / Cauchy integral. A good reference is Dieudonne's Foundations of Modern Analysis, but I think you can also find it in Bourbaki. In any case, it avoids measure theory and gives an ...

1

Assuming the correct regularity on $f$, this is an application of the divergence theorem. Indeed notice that $$\int_{\Omega}f\,dx = \int_{\Omega}\Delta u\, dx = \int_{\partial \Omega}\nabla u \cdot n\, d\mathcal{H}^{N-1} = 0,$$ where the last equality follows from the Neumann condition. This is usually referred to as a compatibility condition.

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